Frank Giustra had been on the Greek island of Lesbos for two hours when the refugee boat came in. Mr. Giustra, the Canadian mining and film magnate whose philanthropic work has yielded close ties to former U.S. president Bill Clinton, had been asked by a friend to consider picking up the tab for a new refugee reception centre.

He had barely set foot on the island last week when he and members of his group spotted flashlight beams along the beach. When they stepped out of their car, they realized that a small boat was travelling through the dark and heading for shore...